Mon. Not. R. Astron. Soc. 415, 1276–1279 (2011) doi:10.1111/j.1365-2966.2011.18778.x

Interaction of p modes with a collection of thin magnetic tubes

R. Jain,1� A. Gascoyne1 and B. W. Hindman2

1Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH2JILA, University of Colorado at Boulder, Boulder, CO 80309-0440, USA

Accepted 2011 March 22. Received 2011 March 18; in original form 2010 September 15

ABSTRACTWe investigate the net effect of a multitude of thin magnetic tubes on the energy of ambientacoustic p modes. A p mode, when incident on a thin magnetic flux tube, excites magneto-hydrodynamic (MHD) tube waves. These tube waves propagate vertically along the flux tubecarrying away energy from the p-mode cavity resulting in the absorption of incident p-modeenergy. We calculate the absorption arising from the excitation of sausage MHD waves withina collection of many non-interacting magnetic flux tubes with differing plasma properties. Wefind that the shape and magnitude of the absorption, when compared with the observationallymeasured absorption, favours a model with a maximum-flux boundary condition applied at thephotosphere and a narrow distribution of plasma β in an ensemble with mean β value between0.5 and 1.

Key words: MHD – Scattering – Sun: helioseismology – Sun: oscillations.

1 IN T RO D U C T I O N

The solar f and p modes of oscillations are stochastically excited inthe solar interior and the measured relationship between their fre-quencies and wavelengths may be used to infer information aboutthe structure and dynamics of the solar interior. It is well estab-lished that these oscillations are influenced by the properties ofmagnetic structures such as sunspots and plages. The studies byBraun, Duvall & LaBonte (1987, 1988), Bogdan & Braun (1995)and Braun & Birch (2008) suggest that both sunspots and plagesare prolific absorbers of p-mode power. Thus, the role of subsurfacefield structures in modifying the properties of f and p modes hasbeen the focus of many theoretical investigations seeking to under-stand the physical mechanism responsible for this absorption (e.g.Spruit 1991; Bogdan & Cally 1995; Bogdan et al. 1996; Crouch& Cally 2005; Jain et al. 2009). In particular, Jain et al. (2009,hereafter referred to as JHBB) modelled a plage as a collectionof thin, untwisted, axisymmetric, vertical magnetic flux tubes andattributed the absorption observed in plage to the excitation of mag-netohydrodynamic (MHD) tube waves by the pummeling of theflux tubes by the p-mode motions. They considered all the fluxtubes comprising the plage to be identical, with the same magneticflux φ and plasma β (the ratio of gas to magnetic pressure). Theyfound that, depending on the boundary condition applied at thephotosphere, absorption coefficients exceeding 50 per cent couldbe achieved. However, in reality, the flux tubes within a plage arenot identical, and thus, in this paper, we consider a heterogeneouscollection of thin flux tubes with varying plasma β. We then in-

�E-mail: R.Jain@sheffield.ac.uk

vestigate the net effect of such a collection on the absorption ofp modes.

2 TH E E QU I L I B R I U M

The thin magnetic flux tubes within the collection are all straight,vertically aligned and axisymmetric. They are embedded within aneutrally stable, polytropically stratified atmosphere. We thus modelthe field-free medium as a plane-parallel atmosphere with constantgravity, g = −g z. The pressure, density and sound speed varywith depth z as power laws with a polytropic index a. We truncatethe polytrope at z = −z0 which represents the model photosphere(see Bogdan et al. 1996 and Hindman & Jain 2008 for the details).Above the truncation depth z > −z0 we assume the existence ofa hot vacuum (ρext → 0 with Text → ∞). For each flux tube inthe collection, we assume that the tube is sufficiently thin that thetube is in thermal equilibrium with its field-free environment. Aconsequence of this assumption is that the plasma β is constantwith depth within the tube and the radial variation of the magneticfield across the tube may be ignored (see Bogdan et al. 1996).However, while the value of β is constant within a given tube, β

is allowed to vary from tube to tube. In particular, we consider theprobability that any given tube has a given value of β is given bythe distribution function,

P (β) = Cβ e− (β−β0)2

2σ2 , β > 0 (1)

where β0 determines the peak of the distribution and σ provides thedistribution’s width. Further, the normalization constant C is givenby

C−1 =∫ ∞

0β e− (β−β0)2

2σ2 dβ. (2)

C© 2011 University of SheffieldMonthly Notices of the Royal Astronomical Society C© 2011 RAS

Interaction of p modes with magnetic tubes 1277

3 TH E G OV E R N I N G E QUAT I O N

The incident p mode in the non-magnetic medium can be describedby a single partial differential equation for the displacement poten-tial �:

∂2�

∂t2= c2∇2� − g

∂�

∂z, (3)

where c2 is the square of the adiabatic sound speed. Writing

μ = (a + 1)

2, ν2 = aω2z0

g, κn = ν2

2knz0,

equation (3) can be transformed into an ordinary differential equa-tion that supports plane wave solutions of the form

�(x, t) = A e−iωteiknxQn(z), (4)

where

Qn(z) = (−2knz)−(μ+1/2)Wκn,μ(−2knz).

Here ω is the temporal frequency and kn(ω) the wavenumber eigen-value; A is the complex wave amplitude and Qn(z) is the verticaleigenfunction proportional to Whittaker’s W function (Abramowitz& Stegun 1964). Assuming that the Lagrangian pressure perturba-tion vanishes at the truncation depth, we calculate the eigenvaluesand eigenfunctions for the truncated polytrope.

In the framework of MHD, thin flux tubes support both longitu-dinal (sausage) waves and transverse (kink) waves but in this paperwe only study the absorption of p modes due to the excitation ofthe sausage waves [see Hindman & Jain (2008) for details of theexcitation of kink waves; see also Spruit (1984) for a derivation ofgoverning equations for tube waves]. When a thin flux tube is pum-meled by a p-mode wave, the vertical fluid displacement of sausagewaves within the tube can be described by[

∂2

∂t2+ 2gz

H

∂2

∂z2+ g(1 + a)

H

∂

∂z

]ξ‖ = (1 + a)(β + 1)

H

∂3�

∂z∂t2,

(5)

where

H = 2a + β(1 + a).

Note that magnetic flux tubes, in general also allow torsionalAlfven waves but we consider a neutrally stable atmosphere forwhich acoustic-gravity waves are irrotational and cannot couple totorsional waves. Furthermore, we do not consider the excitationof tube waves by incident f modes because the absorption mea-surements reported by Braun & Birch (2008) that we are tryingto model lack measurements for the f mode. The observationallymeasured absorption by Braun & Birch (2008) are for p modeswith radial orders 1–4 and show a ‘bell-shaped’ frequency depen-dence; absorption increases with frequency with a peak of roughly20 per cent between 3 and 4 mHz followed by a decrease beyond4 mHz. Azimuthal averaging in these observations results in themeasured absorption coefficients mostly being sensitive to axisym-metric p modes (Aaron Birch, private communication). Hence weare justified in only considering the excitation of sausage waves.

4 A BSORPTION C OEFFICIENT FOR A SINGLEMAGNETIC FLUX TUBE

The observationally measured absorption coefficient is traditionallydefined as the ratio of the difference between the ingoing and out-going power to the ingoing power at the same frequency ω, radialorder n and azimuthal order m. In the current work we assume that

part of the ingoing p-mode energy goes into the excitation of themagnetic tube waves and there is a deficit in the outgoing p-modeenergy as a result of this. We thus, theoretically compute the absorp-tion coefficient, α, associated with this damping mechanism. Theabsorption coefficient due to the excitation of sausage tube wavesis given by (see JHBB for the details of the derivation)

αn = Fn(β, ω)φ ,

Fn(β, ω) = πβ

2(2 + γβ)(β + 1)

(2knz0)a+1

ν2Hn

1

B0z20

× (|� + I∗|2 + |I|2 − |�|2 + S),

B20 = 8π

β + 1P0, (6)

where Hn is the energy flux for an outgoing p mode; B0 is the tube’sphotospheric magnetic field strength, φ is the flux tube’s magneticflux, and P0 is the photospheric pressure in the field-free atmo-sphere. Also, I is the interaction integral between the incident pmode and the sausage tube wave (I∗ is the complex conjugate of I)and � is a boundary condition parameter applied at the photosphere(see Hindman & Jain 2008 for details). The S term is due to thep-mode driver at the surface and is essentially the product ofthe p-mode eigenfunction and the real part of the sausage tubewave displacement.

5 A BSORPTI ON C OEFFI CI ENT FOR AMODELLED PLAG E

Ignoring the excitation of acoustic jacket waves (Bogdan & Cally1995) and the effects of multiple scattering, we model a plage as acollection of many non-interacting flux tubes and estimate the totalabsorption coefficient for the plage as follows. First, we calculatethe absorption due to each magnetic flux tube using the previousexpression. Next, the absorption coefficient αn(r, ω) measured atposition r by local helioseismic techniques such as ridge-filteredholography (Braun & Birch 2008) are related to the tubes througha spatial weighting function or kernel Kn(r, ω),

αn(r, ω) =∑

i

Fn(βi, ω) φi Kn(r i − r, ω), (7)

where each flux tube in the plage is labelled by an index i, and r i ,φi and β i are the position, magnetic flux and plasma β parameter,respectively, for tube i.

If all the flux tubes were identical, the quantity Fn(βi, ω) couldbe taken outside the summation sign and the remaining sum wouldbe simply the kernel-weighted magnetic flux, �n(r, ω), which is ameasurable quantity. This is exactly how the absorption coefficientfor a simulated plage was estimated in JHBB where all flux tubeshad the same β and φ. Here, however, we will be considering aparticular distribution (see equations 1 and 2) of flux tubes. Thus,we compute an ensemble average of the absorption coefficient,

〈αn(r, ω)〉 =⟨∑

i

Fn(βi, ω)φiKn(r i − r, ω)

⟩. (8)

Assuming that the locations r i , magnetic fluxes φi and plasmaparameters β i of the tubes are not correlated, we have

〈αn(r, ω)〉 = Fn(ω)�n(r, ω), (9)

with

Fn(ω) ≡ C∫ ∞

0Fn(β, ω)βe− (β−β0)2

2σ2 dβ , (10)

C© 2011 University of Sheffield, MNRAS 415, 1276–1279Monthly Notices of the Royal Astronomical Society C© 2011 RAS

1278 R. Jain, A. Gascoyne and B. W. Hindman

�n(r, ω) ≡⟨∑

i

φiKn(r i − r, ω)

⟩

≈ ∫dr ′ |B(r ′)| Kn(r ′ − r, ω) . (11)

We show the results for stress-free and maximum-flux boundaryconditions (see Hindman & Jain 2008) satisfied by the sausagewaves at the photosphere. The stress-free boundary condition re-quires the total stress to vanish at the photosphere, and thus, theenergy flux through the surface is zero. The maximum-flux bound-ary condition, on the other hand, is obtained by maximizing theenergy flux that can pass upwards through the model photosphere.Hence, the maximum-flux boundary condition is also a minimalreflection condition where we have an upper limit on the amount ofsausage tube wave energy that can be transmitted into the regionsabove the surface.

6 R ESULTS AND DISCUSSION

We compute equation (9) for different distributions of β values(differing β0 and σ ). The magnetic flux �n and the kernel functionsKn are determined from observation and are identical to those usedby JHBB.

In Fig. 1, we plot the absorption coefficient for a modelled plageas a function of frequency. We have shown the results for maximum-flux (top panel) and stress-free (bottom panels) boundary condi-tions. The symbols in this figure are for a modelled plage that iscomposed of a host of magnetic flux tubes whose β values havebeen drawn from a distribution defined by equations (1) and (2).The three horizontal panels correspond to different β0 values in thedistribution, while the different symbols for the curves denote twodifferent distribution widths σ (0.2 and 2). The solid and dashedcurves correspond to absorption by a plage comprised of a multi-tude of identical flux tubes, all with the same value of β equal tothe mean of the distribution i.e. β = β (see Table 1). As expected,when the distribution of β values is narrow, the absorption coeffi-cient for the distribution of tubes is well matched by a collectionof identical tubes, all with β = β. When the distribution is broad,

Table 1. Mean beta, β for the modelleddistribution function.

β0 0.5 1.0 1.5

β for σ = 0.2 0.58 1 1.5β for σ = 2 2.73 2.98 3.25

substantial differences arise between the distribution of tubes andthe collection of identical tubes for stress-free case. This is becausethe absorption for a single tube is nearly a linear function of β

for the maximum-flux boundary condition while for the stress-freeboundary condition, the absorption is a rapidly decreasing functionof β. This can be seen in Fig. 2 where we have plotted the absorp-tion coefficient, α, for a single tube as a function of plasma β. Thedifferent line-style curves are for various radial order p modes. Thenear-linear dependence of α on β is clear for the maximum-fluxboundary condition case.

Note that for a wide distribution of tubes (σ = 2.0), the absorp-tion coefficient for maximum-flux boundary condition is greaterthan 1 for the p1 and p2 modes.This is clearly not desirable andis a consequence of the weak absorption approximation used inour formalism. We assume that each individual flux tube absorbsenergy from the incident p mode independently of other flux tubesaround it. However, in reality when there are many tubes presentand their net absorption is significant, the ingoing p-mode energysampled by an individual tube is far less than that of the incidentp mode. So, the assumption of weak absorption considered hereoverestimates the absorption. Also, the absorption coefficient issensitive to the upper boundary condition. The stress-free bound-ary condition (complete reflection) produces weak absorption forhigh-β tubes and maximum-flux boundary condition (minimum re-flection) yields very high absorption for high-β tubes. The twoboundary conditions considered here represent lower and upperbounds on the wave reflection at the surface but in reality the ac-tual reflection at the solar surface lies between these two extremecases.

Figure 1. Absorption coefficient as a function of frequency for an ensemble of many thin, magnetic flux tubes with plasma β varying between 0 and ∞. Thetop and bottom panels are for maximum flux and stress-free boundary conditions, respectively, at the photosphere. We chose the distribution function given inequation (1), characterized by the parameters β0 and σ . The filled circles and squares are for σ = 0.2 and 2.0, respectively. The solid and dashed lines representthe corresponding absorption coefficients for a collection of identical tubes in which the β value for each tube was selected to be the same as the mean valueof the distributions defined by equations (1) and (2). Each mode order is denoted by a different colour (online): black (p1), red (p2), green (p3), blue(p4). Notethat absorption decreases with increasing radial order.

C© 2011 University of Sheffield, MNRAS 415, 1276–1279Monthly Notices of the Royal Astronomical Society C© 2011 RAS

Interaction of p modes with magnetic tubes 1279

Figure 2. Absorption coefficient for a single tube as a function of plasma β for a given frequency. Each mode order is denoted by a different linestyle: dashed(p1), dot–dashed (p2), dash–dot–dotted (p3) etc. Note different scale on y-axis.

7 C O N C L U S I O N S

We compute absorption coefficients for a modelled plage assum-ing that the plage is comprised of a collection of non-interacting,vertical, axisymmetric, thin, magnetic-flux tubes. We have shownthat reflection of sausage waves at the solar surface and the plasmaproperties of the magnetic flux tubes in an ensemble are some ofthe factors that strongly affect the energy loss of p modes in amagnetic structure. In particular, we have shown that as long asthe plasma parameter β has a narrow range of variation with β0

< 1, the macroscopic absorption coefficient of the collection ef-fectively depends only on the mean value of β for different betaflux tubes. For a stress-free boundary, the width of the distribu-tion plays an important but lesser role due to the non-linear de-pendence of α on β. In conclusion, the shape and magnitude ofthe absorption, when compared with the observationally measuredabsorption, favours a model with maximum-flux boundary condi-tion and a narrow distribution of plasma β in an ensemble withmean β value between 0.5 and 1. Such conditions produce a collec-tive absorption coefficient with significant amplitude (>0.1) thatreaches a maximum between 3 and 4 mHz. Wide distributions(σ > 1.0) or distributions with β0 > 1 produce absorption coef-ficients that are too large in magnitude. This finding is consistentwith photospheric observations of the spatial distribution of mag-netic field within plage. A range of β values between 0.5 and 1.0corresponds to a range of photospheric field strengths between 1.2and 1.4 kG. Direct observations of the field strength distributionwithin plages typically peak around 1.3 kG with a width of 0.3 kG(Rabin 1992; Martınez Pillet, Lites & Skumanich 1997; Lagg et al.2010).

AC K N OW L E D G M E N T S

RJ and AG acknowledge STFC (UK) for partial fund-ing. BWH acknowledges NASA grants NNG05GM83G,NNG06GD97G, NNX07AH82G, NNX08AJ08G, NNX08AQ28Gand NNX09AB04G.

REFERENCES

Abramowitz M., Stegun I. A., 1964, Handbook of Mathematical Functions.Dover, New York

Bogdan T. J., Braun D. C., 1995, in Hoeksema J. T., ed., Proc. 4th SoHOWorkshop: Helioseismology. ESA-SP 376. ESA Publications Division,Noordwijk, p. 31

Bogdan T. J., Cally P. S., 1995, ApJ, 453, 919Bogdan T. J., Hindman B. W., Cally P. S., Charbonneau P., 1996, ApJ, 465,

406Braun D. C., Birch A. C., 2008, Solar Phys., 251, 267Braun D. C., Duvall T. J., Jr, LaBonte B. J., 1987, ApJ, 319, L27Braun D. C., Duvall T. J., Jr, LaBonte B. J., 1988, ApJ, 335, 1015Crouch A. D., Cally P. S., 2005, Solar Phys., 227, 1Hindman B. W., Jain R., 2008, ApJ, 677, 769Jain R., Hindman B. W., Braun D. C., Birch A. C., 2009, ApJ, 695, 325

(JHBB)Lagg A. et al., 2010, ApJ, 723, L164Martınez Pillet V., Lites B. W., Skumanich A., 1997, ApJ, 474, 810Rabin D., 1992, ApJ, 390, L103Spruit H. C., 1984, in Keil S. L., ed., Small Scale Dynamical Processes in

Quiet Stellar Atmospheres. NSO, Sunspot, p. 249Spruit H. C., 1991, in Toomre J., Gough D. O., eds, Lecture Notes in Physics.

Springer-Verlag, Berlin, p. 388

This paper has been typeset from a TEX/LATEX file prepared by the author.

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